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208 Chapter 11 Incompressible Flow Analysis

11.1 Basic Equations

11.1.1 Differential Equations
The flow behavior in three dimensions is governed by
the full Navier-Stokes equations consisting of the conservation of
mass, momentums and energy. There are five differential equa-
tions which are coupled in complicated form. Solving the full
Navier-Stokes equations requires extensive computational effort.
The equations are thus normally reduced into simplified forms
according to different classes of flow behaviors.
In this chapter, we will concentrate on the steady-state
incompressible laminar flow analysis in two-dimensional Cartesian
coordinates. The Navier-Stokes equations, in this case, consist of
only three differential equations. These equations are: (a) conser-
vation of mass, (b) conservation of momentum in the x-direction,
and (c) conservation of momentum in the y-direction as follows.
u   v   0
x  y 
  p     u     u 



( ) uu  (  ) vu       
x  y  x  x   x   y    y   
  p     v     v 

( ) uv  (  ) vv         
x  y  y  x   x   y    y   
where  is the density, u and v are the velocity components in
the x- and y-directions, respectively, p is the pressure, and  is
the dynamic viscosity.
The three basic unknowns of the three differential
equations above are the velocity components (, ),ux y (, )vx y and
pressure (, )p xy . It is noted that the differential equations form a
set of coupled nonlinear differential equations. Such the set of
differential equations is more difficult to solve as compared to the
differential equations in the preceding chapters.